No abstract

Heckman and Opdam introduced a non-symmetric analogue of Jack polynomials using Cherednik operators. In this paper, we derive a simple recursion formula for these polynomials and formulas relating the symmetric Jack polynomials with the non-symmetric ones. These formulas are then implemented by a closed expression of symmetric and non-symmetric Jack polynomials in terms of certain tableaux. The main application is a proof of a conjecture of Macdonald stating certain integrality and positivity properties of Jack polynomials.Comment: Preprint March 1996, to appear in Invent. Math., 15 pages, Plain Te

No abstract

Abstract. We study generalized and degenerate Whittaker models for reductive groups over local fields of characteristic zero (archimedean or non-archimedean). Our main result is the construction of epimorphisms from the generalized Whittaker model corresponding to a nilpotent orbit to any degenerate Whittaker model corresponding to the same orbit, and to certain degenerate Whittaker models corresponding to bigger orbits. We also give choice-free definitions of generalized and degenerate Whittaker models. Finally, we explain how our methods imply analogous results for Whittaker-Fourier coefficients of automorphic representations.For GLn(F) this implies that a smooth admissible representation π has a generalized Whittaker model WO(π) corresponding to a nilpotent coadjoint orbit O if and only if O lies in the (closure of) the wavefront set WF(π). Previously this was only known to hold for F nonarchimedean and O maximal in WF(π), see [MW87]. We also express WO(π) as an iteration of a version of the Bernstein-Zelevinsky derivatives [BZ77,AGS15a]. This enables us to extend to GLn(R) and GLn(C) several further results from [MW87] on the dimension of WO(π) and on the exactness of the generalized Whittaker functor.

Let G / K be an irreducible Hermitian symmetric space of rank n and let 9 = t + p+ + p_ be the usual decomposition of 9 = Lie(G)c. Let us write P, V, and W = P ® V respectively, for the algebra of holomorphic polynomials, the algebra of constant coefficient holomorphic differential operators, and the "Weyl algebra" of polynomial coefficient holomorphic differential operators on 1'-, and regard all three as K -modules in the usual way.Let I = WK be the algebra of K-invariant differential operators on p_., and let A be the set {oX = (Ab" . ,An) E zn I Ai 2: ... 2: An 2: a}.Then, as we show in the next section, A naturally parametrizes both the irreducible K-submodules P).. of P, as well as a certain distinguished (vector space) basis {D.d of I. The problem we consider is to determine, for A, J.l E A, the scalar eigenvalue cl'(A) by which DI' acts on P>..Our main result, proved in Section 1, is the following characterization of these eigenvalues. For A = (Ai,' .. ,An), let us write IAI for Ai + ... + An; and set Am = {A E A IIAI:::; m}. Also, put P = (Pl,···,Pn) where Pi = d( n -2i + 1)/2 and d is as in the next section.*